
We view \(s{{S}\mathrm {et}_{\hspace{-0.85355pt}\Delta }}= {{C}\mathrm {at}}({\mathbbe {\Delta }}^\mathrm {op}, {{S}\mathrm {et}_{\hspace{-0.85355pt}\Delta }})\)  as endowed with the injective model structure.
Recall that this coincides with the Reedy model structure [1]}.

We let \({{SS}\hspace{-0.56917pt}\mathrm {p}}L_{\mathrm {Sp}}s{{S}\mathrm {et}_{\hspace{-0.85355pt}\Delta }}\)  be the left Bousfield localisation of \(s{{S}\mathrm {et}_{\hspace{-0.85355pt}\Delta }}\)  at the spine inclusions.
This is the model category for Segal spaces.

The model category for complete Segal spaces is the localisation \({CSS}L_{J \boxtimes \Delta ^0} {{SS}\hspace{-0.56917pt}\mathrm {p}}\) .


